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Best approximation by polynomials

Published online by Cambridge University Press:  17 April 2009

Sung Guen Kim
Sung Guen Kim
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu, Korea (702-701), e-mail: [email protected]
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Abstract

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In this paper we show that if E is a separable Banach space, F is a reflexive Banach space, and n, k ∈ ℕ, then every continuous polynomial of degree n from E into F has at least one element of best approximation in the Banach subspace of all continuous k–homogeneous polynomials from E into F.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 2 , October 2003 , pp. 267 - 273
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Alencar, R., Aron, R.M. and Dineen, S., ‘A reflexive space of holomorphic functions in infinitely many variables’, Proc. Amer. Math. Soc. 90 (1984), 407–411.CrossRefGoogle Scholar
[2]Chae, S.B., Holomorphy and calculus in normed spaces (Marcel Dekker, New York, 1985).Google Scholar
[3]Choi, Y.S. and Kim, S.G., ‘Polynomial properties of Banach spaces’, J. Math. Anal. Appl. 190 (1995), 203–210.CrossRefGoogle Scholar
[4]Dineen, S., Complex analysis on infinite dimensional spaces (Springer-Verlag, London, 1999).CrossRefGoogle Scholar
[5]Farmer, J.D., ‘Polynomial reflexivity in Banach spaces’, Israel J. Math. 87 (1994), 257–273.CrossRefGoogle Scholar
[6]Gonzalez, M. and Gutierrez, J.M., ‘Polynomial Grothendieck properties’, Glasgow Math. J. 37 (1995), 211–219.CrossRefGoogle Scholar
[7]Holmes, R.B. and Kripke, B.R., ‘Best approximation by compact operators’, Indiana Univ. Math. J. 21 (1971), 255–263.CrossRefGoogle Scholar
[8]Singer, I., Best approximation in normed linear spaces by elements of linear subspaces (Springer-Verlag, New York, Berlin, 1970).CrossRefGoogle Scholar